A192260 G.f. satisfies: A(x) = Sum_{n>=0} x^n * (1 + A(x))^(2*n) * A(x)^(n^2).

1, 4, 48, 912, 21184, 552320, 15532032, 460947712, 14247537664, 454761822208, 14902431522816, 499315007266816, 17054726818791424, 592541668923539456, 20907267781281054720, 748286964823747526656, 27143591551031801806848, 27143591551031801806848, 997356616630147913089024, 37108619649604340227768320, 1397931208210552892111716352, 53322215792785853528148017152, 2059866344459108561028558880768, 80619871370319975775336625340416

Offset

0,2

Links

Table of n, a(n) for n=0..23.

Formula

Let A = g.f. A(x), then A satisfies:

(1) A = Sum_{n>=0} x^n*(1+A)^(2*n)*A^n * Product_{k=1..n} (1 - x*(1+A)^2*A^(4*k-3))/(1 - x*(1+A)^2*A^(4*k-1))

(2) A = 1/(1- A*(1+A)^2*x/(1- A*(A^2-1)*(1+A)^2*x/(1- A^5*(1+A)^2*x/(1- A^3*(A^4-1)*(1+A)^2*x/(1- A^9*(1+A)^2*x/(1- A^5*(A^6-1)*(1+A)^2*x/(1- A^13*(1+A)^2*x/(1- A^7*(A^8-1)*(1+A)^2*x/(1- ...))))))))) (continued fraction).

The above formulas are due to (1) a q-series identity and (2) a partial elliptic theta function expression.

Example

G.f.: A(x) = 1 + 4*x + 48*x^2 + 912*x^3 + 21184*x^4 + 552320*x^5 +...
Let A = g.f. A(x), then A satisfies:
A = 1 + x*(1+A)^2*A + x^2*(1+A)^4*A^4 + x^3*(1+A)^6*A^9 + x^4*(1+A)^8*A^16 +...
Equivalently,
A = 1 + x*(A + 2*A^2 + A^3) + x^2*(A^4 + 4*A^5 + 6*A^6 + 4*A^7 + A^8) + x^3*(A^9 + 6*A^10 + 15*A^11 + 20*A^12 + 15*A^13 + 6*A^14 + A^15) +...

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*(1+A)^(2*m)*(A+x*O(x^n))^(m^2))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A107595, A192259.

Sequence in context: A167141 A322296 A360484 * A162676 A141119 A052714

Adjacent sequences: A192257 A192258 A192259 * A192261 A192262 A192263

Keyword

nonn

Author

Paul D. Hanna, Jun 26 2011

Status

approved